\(\int \frac {(e+f x)^2 \sinh ^2(c+d x)}{a+i a \sinh (c+d x)} \, dx\) [194]

Optimal result

Integrand size = 31, antiderivative size = 184 \[ \int \frac {(e+f x)^2 \sinh ^2(c+d x)}{a+i a \sinh (c+d x)} \, dx=-\frac {(e+f x)^2}{a d}+\frac {(e+f x)^3}{3 a f}-\frac {2 i f^2 \cosh (c+d x)}{a d^3}-\frac {i (e+f x)^2 \cosh (c+d x)}{a d}+\frac {4 f (e+f x) \log \left (1+i e^{c+d x}\right )}{a d^2}+\frac {4 f^2 \operatorname {PolyLog}\left (2,-i e^{c+d x}\right )}{a d^3}+\frac {2 i f (e+f x) \sinh (c+d x)}{a d^2}-\frac {(e+f x)^2 \tanh \left (\frac {c}{2}+\frac {i \pi }{4}+\frac {d x}{2}\right )}{a d} \] Output:

-(f*x+e)^2/a/d+1/3*(f*x+e)^3/a/f-2*I*f^2*cosh(d*x+c)/a/d^3-I*(f*x+e)^2*cos 
h(d*x+c)/a/d+4*f*(f*x+e)*ln(1+I*exp(d*x+c))/a/d^2+4*f^2*polylog(2,-I*exp(d 
*x+c))/a/d^3+2*I*f*(f*x+e)*sinh(d*x+c)/a/d^2-(f*x+e)^2*tanh(1/2*c+1/4*I*Pi 
+1/2*d*x)/a/d
 

Mathematica [A] (verified)

Time = 1.81 (sec) , antiderivative size = 260, normalized size of antiderivative = 1.41 \[ \int \frac {(e+f x)^2 \sinh ^2(c+d x)}{a+i a \sinh (c+d x)} \, dx=\frac {x \left (3 e^2+3 e f x+f^2 x^2\right )+\frac {6 \left (\frac {d (e+f x) \left (-i d (e+f x)+2 \left (-i+e^c\right ) f \log \left (1-i e^{-c-d x}\right )\right )}{-i+e^c}-2 f^2 \operatorname {PolyLog}\left (2,i e^{-c-d x}\right )\right )}{d^3}-\frac {3 i \cosh (d x) \left (\left (2 f^2+d^2 (e+f x)^2\right ) \cosh (c)-2 d f (e+f x) \sinh (c)\right )}{d^3}-\frac {3 i \left (-2 d f (e+f x) \cosh (c)+\left (2 f^2+d^2 (e+f x)^2\right ) \sinh (c)\right ) \sinh (d x)}{d^3}-\frac {6 (e+f x)^2 \sinh \left (\frac {d x}{2}\right )}{d \left (\cosh \left (\frac {c}{2}\right )+i \sinh \left (\frac {c}{2}\right )\right ) \left (\cosh \left (\frac {1}{2} (c+d x)\right )+i \sinh \left (\frac {1}{2} (c+d x)\right )\right )}}{3 a} \] Input:

Integrate[((e + f*x)^2*Sinh[c + d*x]^2)/(a + I*a*Sinh[c + d*x]),x]
 

Output:

(x*(3*e^2 + 3*e*f*x + f^2*x^2) + (6*((d*(e + f*x)*((-I)*d*(e + f*x) + 2*(- 
I + E^c)*f*Log[1 - I*E^(-c - d*x)]))/(-I + E^c) - 2*f^2*PolyLog[2, I*E^(-c 
 - d*x)]))/d^3 - ((3*I)*Cosh[d*x]*((2*f^2 + d^2*(e + f*x)^2)*Cosh[c] - 2*d 
*f*(e + f*x)*Sinh[c]))/d^3 - ((3*I)*(-2*d*f*(e + f*x)*Cosh[c] + (2*f^2 + d 
^2*(e + f*x)^2)*Sinh[c])*Sinh[d*x])/d^3 - (6*(e + f*x)^2*Sinh[(d*x)/2])/(d 
*(Cosh[c/2] + I*Sinh[c/2])*(Cosh[(c + d*x)/2] + I*Sinh[(c + d*x)/2])))/(3* 
a)
 

Rubi [A] (verified)

Time = 1.61 (sec) , antiderivative size = 204, normalized size of antiderivative = 1.11, number of steps used = 27, number of rules used = 26, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.839, Rules used = {6091, 3042, 26, 3777, 3042, 3777, 26, 3042, 26, 3118, 6091, 17, 3042, 3799, 25, 25, 3042, 4672, 26, 3042, 26, 4199, 26, 2620, 2715, 2838}

Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.

\(\displaystyle \int \frac {(e+f x)^2 \sinh ^2(c+d x)}{a+i a \sinh (c+d x)} \, dx\)

\(\Big \downarrow \) 6091

\(\displaystyle i \int \frac {(e+f x)^2 \sinh (c+d x)}{i \sinh (c+d x) a+a}dx-\frac {i \int (e+f x)^2 \sinh (c+d x)dx}{a}\)

\(\Big \downarrow \) 3042

\(\displaystyle i \int \frac {(e+f x)^2 \sinh (c+d x)}{i \sinh (c+d x) a+a}dx-\frac {i \int -i (e+f x)^2 \sin (i c+i d x)dx}{a}\)

\(\Big \downarrow \) 26

\(\displaystyle i \int \frac {(e+f x)^2 \sinh (c+d x)}{i \sinh (c+d x) a+a}dx-\frac {\int (e+f x)^2 \sin (i c+i d x)dx}{a}\)

\(\Big \downarrow \) 3777

\(\displaystyle i \int \frac {(e+f x)^2 \sinh (c+d x)}{i \sinh (c+d x) a+a}dx-\frac {\frac {i (e+f x)^2 \cosh (c+d x)}{d}-\frac {2 i f \int (e+f x) \cosh (c+d x)dx}{d}}{a}\)

\(\Big \downarrow \) 3042

\(\displaystyle i \int \frac {(e+f x)^2 \sinh (c+d x)}{i \sinh (c+d x) a+a}dx-\frac {\frac {i (e+f x)^2 \cosh (c+d x)}{d}-\frac {2 i f \int (e+f x) \sin \left (i c+i d x+\frac {\pi }{2}\right )dx}{d}}{a}\)

\(\Big \downarrow \) 3777

\(\displaystyle i \int \frac {(e+f x)^2 \sinh (c+d x)}{i \sinh (c+d x) a+a}dx-\frac {\frac {i (e+f x)^2 \cosh (c+d x)}{d}-\frac {2 i f \left (\frac {(e+f x) \sinh (c+d x)}{d}-\frac {i f \int -i \sinh (c+d x)dx}{d}\right )}{d}}{a}\)

\(\Big \downarrow \) 26

\(\displaystyle i \int \frac {(e+f x)^2 \sinh (c+d x)}{i \sinh (c+d x) a+a}dx-\frac {\frac {i (e+f x)^2 \cosh (c+d x)}{d}-\frac {2 i f \left (\frac {(e+f x) \sinh (c+d x)}{d}-\frac {f \int \sinh (c+d x)dx}{d}\right )}{d}}{a}\)

\(\Big \downarrow \) 3042

\(\displaystyle i \int \frac {(e+f x)^2 \sinh (c+d x)}{i \sinh (c+d x) a+a}dx-\frac {\frac {i (e+f x)^2 \cosh (c+d x)}{d}-\frac {2 i f \left (\frac {(e+f x) \sinh (c+d x)}{d}-\frac {f \int -i \sin (i c+i d x)dx}{d}\right )}{d}}{a}\)

\(\Big \downarrow \) 26

\(\displaystyle i \int \frac {(e+f x)^2 \sinh (c+d x)}{i \sinh (c+d x) a+a}dx-\frac {\frac {i (e+f x)^2 \cosh (c+d x)}{d}-\frac {2 i f \left (\frac {(e+f x) \sinh (c+d x)}{d}+\frac {i f \int \sin (i c+i d x)dx}{d}\right )}{d}}{a}\)

\(\Big \downarrow \) 3118

\(\displaystyle i \int \frac {(e+f x)^2 \sinh (c+d x)}{i \sinh (c+d x) a+a}dx-\frac {\frac {i (e+f x)^2 \cosh (c+d x)}{d}-\frac {2 i f \left (\frac {(e+f x) \sinh (c+d x)}{d}-\frac {f \cosh (c+d x)}{d^2}\right )}{d}}{a}\)

\(\Big \downarrow \) 6091

\(\displaystyle i \left (i \int \frac {(e+f x)^2}{i \sinh (c+d x) a+a}dx-\frac {i \int (e+f x)^2dx}{a}\right )-\frac {\frac {i (e+f x)^2 \cosh (c+d x)}{d}-\frac {2 i f \left (\frac {(e+f x) \sinh (c+d x)}{d}-\frac {f \cosh (c+d x)}{d^2}\right )}{d}}{a}\)

\(\Big \downarrow \) 17

\(\displaystyle i \left (i \int \frac {(e+f x)^2}{i \sinh (c+d x) a+a}dx-\frac {i (e+f x)^3}{3 a f}\right )-\frac {\frac {i (e+f x)^2 \cosh (c+d x)}{d}-\frac {2 i f \left (\frac {(e+f x) \sinh (c+d x)}{d}-\frac {f \cosh (c+d x)}{d^2}\right )}{d}}{a}\)

\(\Big \downarrow \) 3042

\(\displaystyle i \left (i \int \frac {(e+f x)^2}{\sin (i c+i d x) a+a}dx-\frac {i (e+f x)^3}{3 a f}\right )-\frac {\frac {i (e+f x)^2 \cosh (c+d x)}{d}-\frac {2 i f \left (\frac {(e+f x) \sinh (c+d x)}{d}-\frac {f \cosh (c+d x)}{d^2}\right )}{d}}{a}\)

\(\Big \downarrow \) 3799

\(\displaystyle i \left (\frac {i \int -(e+f x)^2 \text {csch}^2\left (\frac {c}{2}+\frac {d x}{2}-\frac {i \pi }{4}\right )dx}{2 a}-\frac {i (e+f x)^3}{3 a f}\right )-\frac {\frac {i (e+f x)^2 \cosh (c+d x)}{d}-\frac {2 i f \left (\frac {(e+f x) \sinh (c+d x)}{d}-\frac {f \cosh (c+d x)}{d^2}\right )}{d}}{a}\)

\(\Big \downarrow \) 25

\(\displaystyle i \left (-\frac {i \int -(e+f x)^2 \text {sech}^2\left (\frac {c}{2}+\frac {d x}{2}+\frac {i \pi }{4}\right )dx}{2 a}-\frac {i (e+f x)^3}{3 a f}\right )-\frac {\frac {i (e+f x)^2 \cosh (c+d x)}{d}-\frac {2 i f \left (\frac {(e+f x) \sinh (c+d x)}{d}-\frac {f \cosh (c+d x)}{d^2}\right )}{d}}{a}\)

\(\Big \downarrow \) 25

\(\displaystyle i \left (\frac {i \int (e+f x)^2 \text {sech}^2\left (\frac {c}{2}+\frac {d x}{2}+\frac {i \pi }{4}\right )dx}{2 a}-\frac {i (e+f x)^3}{3 a f}\right )-\frac {\frac {i (e+f x)^2 \cosh (c+d x)}{d}-\frac {2 i f \left (\frac {(e+f x) \sinh (c+d x)}{d}-\frac {f \cosh (c+d x)}{d^2}\right )}{d}}{a}\)

\(\Big \downarrow \) 3042

\(\displaystyle i \left (\frac {i \int (e+f x)^2 \csc \left (\frac {i c}{2}+\frac {i d x}{2}+\frac {\pi }{4}\right )^2dx}{2 a}-\frac {i (e+f x)^3}{3 a f}\right )-\frac {\frac {i (e+f x)^2 \cosh (c+d x)}{d}-\frac {2 i f \left (\frac {(e+f x) \sinh (c+d x)}{d}-\frac {f \cosh (c+d x)}{d^2}\right )}{d}}{a}\)

\(\Big \downarrow \) 4672

\(\displaystyle i \left (\frac {i \left (\frac {2 (e+f x)^2 \tanh \left (\frac {c}{2}+\frac {d x}{2}+\frac {i \pi }{4}\right )}{d}-\frac {4 i f \int -i (e+f x) \tanh \left (\frac {c}{2}+\frac {d x}{2}+\frac {i \pi }{4}\right )dx}{d}\right )}{2 a}-\frac {i (e+f x)^3}{3 a f}\right )-\frac {\frac {i (e+f x)^2 \cosh (c+d x)}{d}-\frac {2 i f \left (\frac {(e+f x) \sinh (c+d x)}{d}-\frac {f \cosh (c+d x)}{d^2}\right )}{d}}{a}\)

\(\Big \downarrow \) 26

\(\displaystyle i \left (\frac {i \left (\frac {2 (e+f x)^2 \tanh \left (\frac {c}{2}+\frac {d x}{2}+\frac {i \pi }{4}\right )}{d}-\frac {4 f \int (e+f x) \tanh \left (\frac {c}{2}+\frac {d x}{2}+\frac {i \pi }{4}\right )dx}{d}\right )}{2 a}-\frac {i (e+f x)^3}{3 a f}\right )-\frac {\frac {i (e+f x)^2 \cosh (c+d x)}{d}-\frac {2 i f \left (\frac {(e+f x) \sinh (c+d x)}{d}-\frac {f \cosh (c+d x)}{d^2}\right )}{d}}{a}\)

\(\Big \downarrow \) 3042

\(\displaystyle i \left (\frac {i \left (\frac {2 (e+f x)^2 \tanh \left (\frac {c}{2}+\frac {d x}{2}+\frac {i \pi }{4}\right )}{d}-\frac {4 f \int -i (e+f x) \tan \left (\frac {i c}{2}+\frac {i d x}{2}-\frac {\pi }{4}\right )dx}{d}\right )}{2 a}-\frac {i (e+f x)^3}{3 a f}\right )-\frac {\frac {i (e+f x)^2 \cosh (c+d x)}{d}-\frac {2 i f \left (\frac {(e+f x) \sinh (c+d x)}{d}-\frac {f \cosh (c+d x)}{d^2}\right )}{d}}{a}\)

\(\Big \downarrow \) 26

\(\displaystyle i \left (\frac {i \left (\frac {4 i f \int (e+f x) \tan \left (\frac {i c}{2}+\frac {i d x}{2}-\frac {\pi }{4}\right )dx}{d}+\frac {2 (e+f x)^2 \tanh \left (\frac {c}{2}+\frac {d x}{2}+\frac {i \pi }{4}\right )}{d}\right )}{2 a}-\frac {i (e+f x)^3}{3 a f}\right )-\frac {\frac {i (e+f x)^2 \cosh (c+d x)}{d}-\frac {2 i f \left (\frac {(e+f x) \sinh (c+d x)}{d}-\frac {f \cosh (c+d x)}{d^2}\right )}{d}}{a}\)

\(\Big \downarrow \) 4199

\(\displaystyle i \left (\frac {i \left (\frac {4 i f \left (2 i \int \frac {i e^{c+d x} (e+f x)}{1+i e^{c+d x}}dx-\frac {i (e+f x)^2}{2 f}\right )}{d}+\frac {2 (e+f x)^2 \tanh \left (\frac {c}{2}+\frac {d x}{2}+\frac {i \pi }{4}\right )}{d}\right )}{2 a}-\frac {i (e+f x)^3}{3 a f}\right )-\frac {\frac {i (e+f x)^2 \cosh (c+d x)}{d}-\frac {2 i f \left (\frac {(e+f x) \sinh (c+d x)}{d}-\frac {f \cosh (c+d x)}{d^2}\right )}{d}}{a}\)

\(\Big \downarrow \) 26

\(\displaystyle i \left (\frac {i \left (\frac {4 i f \left (-2 \int \frac {e^{c+d x} (e+f x)}{1+i e^{c+d x}}dx-\frac {i (e+f x)^2}{2 f}\right )}{d}+\frac {2 (e+f x)^2 \tanh \left (\frac {c}{2}+\frac {d x}{2}+\frac {i \pi }{4}\right )}{d}\right )}{2 a}-\frac {i (e+f x)^3}{3 a f}\right )-\frac {\frac {i (e+f x)^2 \cosh (c+d x)}{d}-\frac {2 i f \left (\frac {(e+f x) \sinh (c+d x)}{d}-\frac {f \cosh (c+d x)}{d^2}\right )}{d}}{a}\)

\(\Big \downarrow \) 2620

\(\displaystyle i \left (\frac {i \left (\frac {4 i f \left (-2 \left (\frac {i f \int \log \left (1+i e^{c+d x}\right )dx}{d}-\frac {i (e+f x) \log \left (1+i e^{c+d x}\right )}{d}\right )-\frac {i (e+f x)^2}{2 f}\right )}{d}+\frac {2 (e+f x)^2 \tanh \left (\frac {c}{2}+\frac {d x}{2}+\frac {i \pi }{4}\right )}{d}\right )}{2 a}-\frac {i (e+f x)^3}{3 a f}\right )-\frac {\frac {i (e+f x)^2 \cosh (c+d x)}{d}-\frac {2 i f \left (\frac {(e+f x) \sinh (c+d x)}{d}-\frac {f \cosh (c+d x)}{d^2}\right )}{d}}{a}\)

\(\Big \downarrow \) 2715

\(\displaystyle i \left (\frac {i \left (\frac {4 i f \left (-2 \left (\frac {i f \int e^{-c-d x} \log \left (1+i e^{c+d x}\right )de^{c+d x}}{d^2}-\frac {i (e+f x) \log \left (1+i e^{c+d x}\right )}{d}\right )-\frac {i (e+f x)^2}{2 f}\right )}{d}+\frac {2 (e+f x)^2 \tanh \left (\frac {c}{2}+\frac {d x}{2}+\frac {i \pi }{4}\right )}{d}\right )}{2 a}-\frac {i (e+f x)^3}{3 a f}\right )-\frac {\frac {i (e+f x)^2 \cosh (c+d x)}{d}-\frac {2 i f \left (\frac {(e+f x) \sinh (c+d x)}{d}-\frac {f \cosh (c+d x)}{d^2}\right )}{d}}{a}\)

\(\Big \downarrow \) 2838

\(\displaystyle i \left (\frac {i \left (\frac {4 i f \left (-2 \left (-\frac {i f \operatorname {PolyLog}\left (2,-i e^{c+d x}\right )}{d^2}-\frac {i (e+f x) \log \left (1+i e^{c+d x}\right )}{d}\right )-\frac {i (e+f x)^2}{2 f}\right )}{d}+\frac {2 (e+f x)^2 \tanh \left (\frac {c}{2}+\frac {d x}{2}+\frac {i \pi }{4}\right )}{d}\right )}{2 a}-\frac {i (e+f x)^3}{3 a f}\right )-\frac {\frac {i (e+f x)^2 \cosh (c+d x)}{d}-\frac {2 i f \left (\frac {(e+f x) \sinh (c+d x)}{d}-\frac {f \cosh (c+d x)}{d^2}\right )}{d}}{a}\)

Input:

Int[((e + f*x)^2*Sinh[c + d*x]^2)/(a + I*a*Sinh[c + d*x]),x]
 

Output:

-(((I*(e + f*x)^2*Cosh[c + d*x])/d - ((2*I)*f*(-((f*Cosh[c + d*x])/d^2) + 
((e + f*x)*Sinh[c + d*x])/d))/d)/a) + I*(((-1/3*I)*(e + f*x)^3)/(a*f) + (( 
I/2)*(((4*I)*f*(((-1/2*I)*(e + f*x)^2)/f - 2*(((-I)*(e + f*x)*Log[1 + I*E^ 
(c + d*x)])/d - (I*f*PolyLog[2, (-I)*E^(c + d*x)])/d^2)))/d + (2*(e + f*x) 
^2*Tanh[c/2 + (I/4)*Pi + (d*x)/2])/d))/a)
 

Defintions of rubi rules used

rule 17

Int[(c_.)*((a_.) + (b_.)*(x_))^(m_.), x_Symbol] :> Simp[c*((a + b*x)^(m + 1 
)/(b*(m + 1))), x] /; FreeQ[{a, b, c, m}, x] && NeQ[m, -1]
 

rule 25

Int[-(Fx_), x_Symbol] :> Simp[Identity[-1]   Int[Fx, x], x]
 

rule 26

Int[(Complex[0, a_])*(Fx_), x_Symbol] :> Simp[(Complex[Identity[0], a])   I 
nt[Fx, x], x] /; FreeQ[a, x] && EqQ[a^2, 1]
 

rule 2620

Int[(((F_)^((g_.)*((e_.) + (f_.)*(x_))))^(n_.)*((c_.) + (d_.)*(x_))^(m_.))/ 
((a_) + (b_.)*((F_)^((g_.)*((e_.) + (f_.)*(x_))))^(n_.)), x_Symbol] :> Simp 
[((c + d*x)^m/(b*f*g*n*Log[F]))*Log[1 + b*((F^(g*(e + f*x)))^n/a)], x] - Si 
mp[d*(m/(b*f*g*n*Log[F]))   Int[(c + d*x)^(m - 1)*Log[1 + b*((F^(g*(e + f*x 
)))^n/a)], x], x] /; FreeQ[{F, a, b, c, d, e, f, g, n}, x] && IGtQ[m, 0]
 

rule 2715

Int[Log[(a_) + (b_.)*((F_)^((e_.)*((c_.) + (d_.)*(x_))))^(n_.)], x_Symbol] 
:> Simp[1/(d*e*n*Log[F])   Subst[Int[Log[a + b*x]/x, x], x, (F^(e*(c + d*x) 
))^n], x] /; FreeQ[{F, a, b, c, d, e, n}, x] && GtQ[a, 0]
 

rule 2838

Int[Log[(c_.)*((d_) + (e_.)*(x_)^(n_.))]/(x_), x_Symbol] :> Simp[-PolyLog[2 
, (-c)*e*x^n]/n, x] /; FreeQ[{c, d, e, n}, x] && EqQ[c*d, 1]
 

rule 3042

Int[u_, x_Symbol] :> Int[DeactivateTrig[u, x], x] /; FunctionOfTrigOfLinear 
Q[u, x]
 

rule 3118

Int[sin[(c_.) + (d_.)*(x_)], x_Symbol] :> Simp[-Cos[c + d*x]/d, x] /; FreeQ 
[{c, d}, x]
 

rule 3777

Int[((c_.) + (d_.)*(x_))^(m_.)*sin[(e_.) + (f_.)*(x_)], x_Symbol] :> Simp[( 
-(c + d*x)^m)*(Cos[e + f*x]/f), x] + Simp[d*(m/f)   Int[(c + d*x)^(m - 1)*C 
os[e + f*x], x], x] /; FreeQ[{c, d, e, f}, x] && GtQ[m, 0]
 

rule 3799

Int[((c_.) + (d_.)*(x_))^(m_.)*((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(n_.) 
, x_Symbol] :> Simp[(2*a)^n   Int[(c + d*x)^m*Sin[(1/2)*(e + Pi*(a/(2*b))) 
+ f*(x/2)]^(2*n), x], x] /; FreeQ[{a, b, c, d, e, f, m}, x] && EqQ[a^2 - b^ 
2, 0] && IntegerQ[n] && (GtQ[n, 0] || IGtQ[m, 0])
 

rule 4199

Int[((c_.) + (d_.)*(x_))^(m_.)*tan[(e_.) + Pi*(k_.) + (Complex[0, fz_])*(f_ 
.)*(x_)], x_Symbol] :> Simp[(-I)*((c + d*x)^(m + 1)/(d*(m + 1))), x] + Simp 
[2*I   Int[((c + d*x)^m*(E^(2*((-I)*e + f*fz*x))/(1 + E^(2*((-I)*e + f*fz*x 
))/E^(2*I*k*Pi))))/E^(2*I*k*Pi), x], x] /; FreeQ[{c, d, e, f, fz}, x] && In 
tegerQ[4*k] && IGtQ[m, 0]
 

rule 4672

Int[csc[(e_.) + (f_.)*(x_)]^2*((c_.) + (d_.)*(x_))^(m_.), x_Symbol] :> Simp 
[(-(c + d*x)^m)*(Cot[e + f*x]/f), x] + Simp[d*(m/f)   Int[(c + d*x)^(m - 1) 
*Cot[e + f*x], x], x] /; FreeQ[{c, d, e, f}, x] && GtQ[m, 0]
 

rule 6091

Int[(((e_.) + (f_.)*(x_))^(m_.)*Sinh[(c_.) + (d_.)*(x_)]^(n_.))/((a_) + (b_ 
.)*Sinh[(c_.) + (d_.)*(x_)]), x_Symbol] :> Simp[1/b   Int[(e + f*x)^m*Sinh[ 
c + d*x]^(n - 1), x], x] - Simp[a/b   Int[(e + f*x)^m*(Sinh[c + d*x]^(n - 1 
)/(a + b*Sinh[c + d*x])), x], x] /; FreeQ[{a, b, c, d, e, f}, x] && IGtQ[m, 
 0] && IGtQ[n, 0]
 

Maple [B] (verified)

Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 384 vs. \(2 (168 ) = 336\).

Time = 0.65 (sec) , antiderivative size = 385, normalized size of antiderivative = 2.09

method	result	size
risch	\(\frac {f^{2} x^{3}}{3 a}+\frac {f e \,x^{2}}{a}+\frac {e^{2} x}{a}+\frac {e^{3}}{3 a f}-\frac {i \left (d^{2} x^{2} f^{2}+2 d^{2} e f x +d^{2} e^{2}-2 d \,f^{2} x -2 d e f +2 f^{2}\right ) {\mathrm e}^{d x +c}}{2 d^{3} a}-\frac {i \left (d^{2} x^{2} f^{2}+2 d^{2} e f x +d^{2} e^{2}+2 d \,f^{2} x +2 d e f +2 f^{2}\right ) {\mathrm e}^{-d x -c}}{2 d^{3} a}-\frac {2 i \left (x^{2} f^{2}+2 e f x +e^{2}\right )}{d a \left ({\mathrm e}^{d x +c}-i\right )}+\frac {4 f \ln \left ({\mathrm e}^{d x +c}-i\right ) e}{a \,d^{2}}-\frac {4 f \ln \left ({\mathrm e}^{d x +c}\right ) e}{a \,d^{2}}-\frac {2 f^{2} x^{2}}{a d}-\frac {4 f^{2} c x}{a \,d^{2}}-\frac {2 f^{2} c^{2}}{a \,d^{3}}+\frac {4 f^{2} \ln \left (1+i {\mathrm e}^{d x +c}\right ) x}{a \,d^{2}}+\frac {4 f^{2} \ln \left (1+i {\mathrm e}^{d x +c}\right ) c}{a \,d^{3}}+\frac {4 f^{2} \operatorname {polylog}\left (2, -i {\mathrm e}^{d x +c}\right )}{a \,d^{3}}-\frac {4 f^{2} c \ln \left ({\mathrm e}^{d x +c}-i\right )}{a \,d^{3}}+\frac {4 f^{2} c \ln \left ({\mathrm e}^{d x +c}\right )}{a \,d^{3}}\)	\(385\)

Input:

int((f*x+e)^2*sinh(d*x+c)^2/(a+I*a*sinh(d*x+c)),x,method=_RETURNVERBOSE)
 

Output:

1/3/a*f^2*x^3+1/a*f*e*x^2+1/a*e^2*x+1/3/a/f*e^3-1/2*I*(d^2*f^2*x^2+2*d^2*e 
*f*x+d^2*e^2-2*d*f^2*x-2*d*e*f+2*f^2)/d^3/a*exp(d*x+c)-1/2*I*(d^2*f^2*x^2+ 
2*d^2*e*f*x+d^2*e^2+2*d*f^2*x+2*d*e*f+2*f^2)/d^3/a*exp(-d*x-c)-2*I*(f^2*x^ 
2+2*e*f*x+e^2)/d/a/(exp(d*x+c)-I)+4/a/d^2*f*ln(exp(d*x+c)-I)*e-4/a/d^2*f*l 
n(exp(d*x+c))*e-2/a/d*f^2*x^2-4/a/d^2*f^2*c*x-2/a/d^3*f^2*c^2+4/a/d^2*f^2* 
ln(1+I*exp(d*x+c))*x+4/a/d^3*f^2*ln(1+I*exp(d*x+c))*c+4*f^2*polylog(2,-I*e 
xp(d*x+c))/a/d^3-4/a/d^3*f^2*c*ln(exp(d*x+c)-I)+4/a/d^3*f^2*c*ln(exp(d*x+c 
))
 

Fricas [B] (verification not implemented)

Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 474 vs. \(2 (161) = 322\).

Time = 0.12 (sec) , antiderivative size = 474, normalized size of antiderivative = 2.58 \[ \int \frac {(e+f x)^2 \sinh ^2(c+d x)}{a+i a \sinh (c+d x)} \, dx=-\frac {3 \, d^{2} f^{2} x^{2} + 3 \, d^{2} e^{2} + 6 \, d e f + 6 \, f^{2} + 6 \, {\left (d^{2} e f + d f^{2}\right )} x - 24 \, {\left (f^{2} e^{\left (2 \, d x + 2 \, c\right )} - i \, f^{2} e^{\left (d x + c\right )}\right )} {\rm Li}_2\left (-i \, e^{\left (d x + c\right )}\right ) + 3 \, {\left (i \, d^{2} f^{2} x^{2} + i \, d^{2} e^{2} - 2 i \, d e f + 2 i \, f^{2} + 2 \, {\left (i \, d^{2} e f - i \, d f^{2}\right )} x\right )} e^{\left (3 \, d x + 3 \, c\right )} - {\left (2 \, d^{3} f^{2} x^{3} - 3 \, d^{2} e^{2} - 6 \, {\left (4 \, c - 1\right )} d e f + 6 \, {\left (2 \, c^{2} - 1\right )} f^{2} + 3 \, {\left (2 \, d^{3} e f - 5 \, d^{2} f^{2}\right )} x^{2} + 6 \, {\left (d^{3} e^{2} - 5 \, d^{2} e f + d f^{2}\right )} x\right )} e^{\left (2 \, d x + 2 \, c\right )} - {\left (-2 i \, d^{3} f^{2} x^{3} - 15 i \, d^{2} e^{2} - 6 \, {\left (-4 i \, c + i\right )} d e f - 6 \, {\left (2 i \, c^{2} + i\right )} f^{2} - 3 \, {\left (2 i \, d^{3} e f + i \, d^{2} f^{2}\right )} x^{2} - 6 \, {\left (i \, d^{3} e^{2} + i \, d^{2} e f + i \, d f^{2}\right )} x\right )} e^{\left (d x + c\right )} - 24 \, {\left ({\left (d e f - c f^{2}\right )} e^{\left (2 \, d x + 2 \, c\right )} - {\left (i \, d e f - i \, c f^{2}\right )} e^{\left (d x + c\right )}\right )} \log \left (e^{\left (d x + c\right )} - i\right ) - 24 \, {\left ({\left (d f^{2} x + c f^{2}\right )} e^{\left (2 \, d x + 2 \, c\right )} - {\left (i \, d f^{2} x + i \, c f^{2}\right )} e^{\left (d x + c\right )}\right )} \log \left (i \, e^{\left (d x + c\right )} + 1\right )}{6 \, {\left (a d^{3} e^{\left (2 \, d x + 2 \, c\right )} - i \, a d^{3} e^{\left (d x + c\right )}\right )}} \] Input:

integrate((f*x+e)^2*sinh(d*x+c)^2/(a+I*a*sinh(d*x+c)),x, algorithm="fricas 
")
 

Output:

-1/6*(3*d^2*f^2*x^2 + 3*d^2*e^2 + 6*d*e*f + 6*f^2 + 6*(d^2*e*f + d*f^2)*x 
- 24*(f^2*e^(2*d*x + 2*c) - I*f^2*e^(d*x + c))*dilog(-I*e^(d*x + c)) + 3*( 
I*d^2*f^2*x^2 + I*d^2*e^2 - 2*I*d*e*f + 2*I*f^2 + 2*(I*d^2*e*f - I*d*f^2)* 
x)*e^(3*d*x + 3*c) - (2*d^3*f^2*x^3 - 3*d^2*e^2 - 6*(4*c - 1)*d*e*f + 6*(2 
*c^2 - 1)*f^2 + 3*(2*d^3*e*f - 5*d^2*f^2)*x^2 + 6*(d^3*e^2 - 5*d^2*e*f + d 
*f^2)*x)*e^(2*d*x + 2*c) - (-2*I*d^3*f^2*x^3 - 15*I*d^2*e^2 - 6*(-4*I*c + 
I)*d*e*f - 6*(2*I*c^2 + I)*f^2 - 3*(2*I*d^3*e*f + I*d^2*f^2)*x^2 - 6*(I*d^ 
3*e^2 + I*d^2*e*f + I*d*f^2)*x)*e^(d*x + c) - 24*((d*e*f - c*f^2)*e^(2*d*x 
 + 2*c) - (I*d*e*f - I*c*f^2)*e^(d*x + c))*log(e^(d*x + c) - I) - 24*((d*f 
^2*x + c*f^2)*e^(2*d*x + 2*c) - (I*d*f^2*x + I*c*f^2)*e^(d*x + c))*log(I*e 
^(d*x + c) + 1))/(a*d^3*e^(2*d*x + 2*c) - I*a*d^3*e^(d*x + c))
 

Sympy [F]

\[ \int \frac {(e+f x)^2 \sinh ^2(c+d x)}{a+i a \sinh (c+d x)} \, dx=\frac {- 2 i e^{2} - 4 i e f x - 2 i f^{2} x^{2}}{a d e^{c} e^{d x} - i a d} - \frac {i \left (\int \frac {i d e^{2}}{e^{c} e^{2 d x} - i e^{d x}}\, dx + \int \frac {i d f^{2} x^{2}}{e^{c} e^{2 d x} - i e^{d x}}\, dx + \int \frac {d e^{2} e^{c} e^{d x}}{e^{c} e^{2 d x} - i e^{d x}}\, dx + \int \frac {d e^{2} e^{3 c} e^{3 d x}}{e^{c} e^{2 d x} - i e^{d x}}\, dx + \int \left (- \frac {8 e f e^{c} e^{d x}}{e^{c} e^{2 d x} - i e^{d x}}\right )\, dx + \int \left (- \frac {8 f^{2} x e^{c} e^{d x}}{e^{c} e^{2 d x} - i e^{d x}}\right )\, dx + \int \frac {2 i d e f x}{e^{c} e^{2 d x} - i e^{d x}}\, dx + \int \frac {i d e^{2} e^{2 c} e^{2 d x}}{e^{c} e^{2 d x} - i e^{d x}}\, dx + \int \frac {d f^{2} x^{2} e^{c} e^{d x}}{e^{c} e^{2 d x} - i e^{d x}}\, dx + \int \frac {d f^{2} x^{2} e^{3 c} e^{3 d x}}{e^{c} e^{2 d x} - i e^{d x}}\, dx + \int \frac {i d f^{2} x^{2} e^{2 c} e^{2 d x}}{e^{c} e^{2 d x} - i e^{d x}}\, dx + \int \frac {2 d e f x e^{c} e^{d x}}{e^{c} e^{2 d x} - i e^{d x}}\, dx + \int \frac {2 d e f x e^{3 c} e^{3 d x}}{e^{c} e^{2 d x} - i e^{d x}}\, dx + \int \frac {2 i d e f x e^{2 c} e^{2 d x}}{e^{c} e^{2 d x} - i e^{d x}}\, dx\right ) e^{- c}}{2 a d} \] Input:

integrate((f*x+e)**2*sinh(d*x+c)**2/(a+I*a*sinh(d*x+c)),x)
 

Output:

(-2*I*e**2 - 4*I*e*f*x - 2*I*f**2*x**2)/(a*d*exp(c)*exp(d*x) - I*a*d) - I* 
(Integral(I*d*e**2/(exp(c)*exp(2*d*x) - I*exp(d*x)), x) + Integral(I*d*f** 
2*x**2/(exp(c)*exp(2*d*x) - I*exp(d*x)), x) + Integral(d*e**2*exp(c)*exp(d 
*x)/(exp(c)*exp(2*d*x) - I*exp(d*x)), x) + Integral(d*e**2*exp(3*c)*exp(3* 
d*x)/(exp(c)*exp(2*d*x) - I*exp(d*x)), x) + Integral(-8*e*f*exp(c)*exp(d*x 
)/(exp(c)*exp(2*d*x) - I*exp(d*x)), x) + Integral(-8*f**2*x*exp(c)*exp(d*x 
)/(exp(c)*exp(2*d*x) - I*exp(d*x)), x) + Integral(2*I*d*e*f*x/(exp(c)*exp( 
2*d*x) - I*exp(d*x)), x) + Integral(I*d*e**2*exp(2*c)*exp(2*d*x)/(exp(c)*e 
xp(2*d*x) - I*exp(d*x)), x) + Integral(d*f**2*x**2*exp(c)*exp(d*x)/(exp(c) 
*exp(2*d*x) - I*exp(d*x)), x) + Integral(d*f**2*x**2*exp(3*c)*exp(3*d*x)/( 
exp(c)*exp(2*d*x) - I*exp(d*x)), x) + Integral(I*d*f**2*x**2*exp(2*c)*exp( 
2*d*x)/(exp(c)*exp(2*d*x) - I*exp(d*x)), x) + Integral(2*d*e*f*x*exp(c)*ex 
p(d*x)/(exp(c)*exp(2*d*x) - I*exp(d*x)), x) + Integral(2*d*e*f*x*exp(3*c)* 
exp(3*d*x)/(exp(c)*exp(2*d*x) - I*exp(d*x)), x) + Integral(2*I*d*e*f*x*exp 
(2*c)*exp(2*d*x)/(exp(c)*exp(2*d*x) - I*exp(d*x)), x))*exp(-c)/(2*a*d)
 

Maxima [F]

\[ \int \frac {(e+f x)^2 \sinh ^2(c+d x)}{a+i a \sinh (c+d x)} \, dx=\int { \frac {{\left (f x + e\right )}^{2} \sinh \left (d x + c\right )^{2}}{i \, a \sinh \left (d x + c\right ) + a} \,d x } \] Input:

integrate((f*x+e)^2*sinh(d*x+c)^2/(a+I*a*sinh(d*x+c)),x, algorithm="maxima 
")
 

Output:

-e*f*(2*x*e^(d*x + c)/(a*d*e^(d*x + c) - I*a*d) + (I*d^2*x^2*e^c + I*d*x*e 
^c - (-I*d*x*e^(3*c) + I*e^(3*c))*e^(2*d*x) - (d^2*x^2*e^(2*c) - 3*d*x*e^( 
2*c) + e^(2*c))*e^(d*x) + (d*x + 1)*e^(-d*x) + I*e^c)/(a*d^2*e^(d*x + 2*c) 
 - I*a*d^2*e^c) - 4*log((e^(d*x + c) - I)*e^(-c))/(a*d^2)) - 1/6*f^2*((2*I 
*d^3*x^3 + 15*I*d^2*x^2 + 6*I*d*x - 3*(-I*d^2*x^2*e^(2*c) + 2*I*d*x*e^(2*c 
) - 2*I*e^(2*c))*e^(2*d*x) - (2*d^3*x^3*e^c - 3*d^2*x^2*e^c + 6*d*x*e^c - 
6*e^c)*e^(d*x) + 6*I)/(a*d^3*e^(d*x + c) - I*a*d^3) - 24*I*integrate(x/(a* 
d*e^(d*x + c) - I*a*d), x)) + 1/2*e^2*(2*(d*x + c)/(a*d) + (-5*I*e^(-d*x - 
 c) + 1)/((I*a*e^(-d*x - c) + a*e^(-2*d*x - 2*c))*d) - I*e^(-d*x - c)/(a*d 
))
 

Giac [F]

\[ \int \frac {(e+f x)^2 \sinh ^2(c+d x)}{a+i a \sinh (c+d x)} \, dx=\int { \frac {{\left (f x + e\right )}^{2} \sinh \left (d x + c\right )^{2}}{i \, a \sinh \left (d x + c\right ) + a} \,d x } \] Input:

integrate((f*x+e)^2*sinh(d*x+c)^2/(a+I*a*sinh(d*x+c)),x, algorithm="giac")
 

Output:

integrate((f*x + e)^2*sinh(d*x + c)^2/(I*a*sinh(d*x + c) + a), x)
 

Mupad [F(-1)]

Timed out. \[ \int \frac {(e+f x)^2 \sinh ^2(c+d x)}{a+i a \sinh (c+d x)} \, dx=\int \frac {{\mathrm {sinh}\left (c+d\,x\right )}^2\,{\left (e+f\,x\right )}^2}{a+a\,\mathrm {sinh}\left (c+d\,x\right )\,1{}\mathrm {i}} \,d x \] Input:

int((sinh(c + d*x)^2*(e + f*x)^2)/(a + a*sinh(c + d*x)*1i),x)
 

Output:

int((sinh(c + d*x)^2*(e + f*x)^2)/(a + a*sinh(c + d*x)*1i), x)
 

Reduce [F]

\[ \int \frac {(e+f x)^2 \sinh ^2(c+d x)}{a+i a \sinh (c+d x)} \, dx=\frac {-3 \cosh \left (d x +c \right ) d^{2} e^{2} i -6 \cosh \left (d x +c \right ) d^{2} e f i x -3 \cosh \left (d x +c \right ) d^{2} f^{2} i \,x^{2}-6 \cosh \left (d x +c \right ) f^{2} i -3 \left (\int \frac {x^{2}}{\sinh \left (d x +c \right ) i +1}d x \right ) d^{3} f^{2}-6 \left (\int \frac {x}{\sinh \left (d x +c \right ) i +1}d x \right ) d^{3} e f -3 \left (\int \frac {1}{\sinh \left (d x +c \right ) i +1}d x \right ) d^{3} e^{2}+6 \sinh \left (d x +c \right ) d e f i +6 \sinh \left (d x +c \right ) d \,f^{2} i x +3 d^{3} e^{2} x +3 d^{3} e f \,x^{2}+d^{3} f^{2} x^{3}}{3 a \,d^{3}} \] Input:

int((f*x+e)^2*sinh(d*x+c)^2/(a+I*a*sinh(d*x+c)),x)
 

Output:

( - 3*cosh(c + d*x)*d**2*e**2*i - 6*cosh(c + d*x)*d**2*e*f*i*x - 3*cosh(c 
+ d*x)*d**2*f**2*i*x**2 - 6*cosh(c + d*x)*f**2*i - 3*int(x**2/(sinh(c + d* 
x)*i + 1),x)*d**3*f**2 - 6*int(x/(sinh(c + d*x)*i + 1),x)*d**3*e*f - 3*int 
(1/(sinh(c + d*x)*i + 1),x)*d**3*e**2 + 6*sinh(c + d*x)*d*e*f*i + 6*sinh(c 
 + d*x)*d*f**2*i*x + 3*d**3*e**2*x + 3*d**3*e*f*x**2 + d**3*f**2*x**3)/(3* 
a*d**3)